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Math Help - differential analysis

  1. #1
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    differential analysis

    Find an example of a differentiable function
    f : R R such that f'(0) = 1 > 0 but f is not monotonic increasing on any interval (0,delta).
    Suppose that
    f : R R has derivatives of all orders. Prove that the same

    is true of
    F(x) := exp (f(x)).

    not sure on either of these two questions..

    thanks in advance
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  2. #2
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by James0502 View Post
    Find an example of a differentiable function
    f : R R such that f'(0) = 1 > 0 but f is not monotonic increasing on any interval (0,delta).
    Suppose that
    f : R R has derivatives of all orders. Prove that the same

    is true of
    F(x) := exp (f(x)).

    not sure on either of these two questions..

    thanks in advance
    For the first one, the following link might be helpful Weierstrass function - Wikipedia, the free encyclopedia, and for the second one, you may refer to http://www.mathhelpforum.com/math-he...-new-post.html
    About the first one I really wonder if there is an explicit answer, as I remember such a discussion was made previously .
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  3. #3
    MHF Contributor

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    Quote Originally Posted by James0502 View Post
    Find an example of a differentiable function
    f : R R such that f'(0) = 1 > 0 but f is not monotonic increasing on any interval (0,delta).
    Suppose that
    f : R R has derivatives of all orders. Prove that the same

    is true of
    F(x) := exp (f(x)).

    not sure on either of these two questions..

    thanks in advance
    For the first question, you need f to be differentiable everywhere, but f' must be discontinuous at 0 (otherwise, it would be positive near 0 and f would be increasing).

    A quite famous example of a differentiable function which is not continuously differentiable at 0 is the following: g(x)=x^2\sin\frac{1}{x} with g(0)=0. However, g'(0)=0, so we should change it a little... You can prove that the function f:x\mapsto f(x)=x + 2x^2\sin\frac{1}{x} with f(0)=0 is an answer to your problem. The "2" is probably optional but may simplify the proof.

    You have to prove that f is differentiable on (0,+\infty) (straightforward), continuous at 0, differentiable at 0 (look at \frac{f(x)}{x}), that f'(0)=1 and that f'(x) has not a constant sign on any neighbourhood (0,h) of 0.
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